Effective electrical manipulation of a topological antiferromagnet by orbital torques

The electrical control of the non-trivial topology in Weyl antiferromagnets is of great interest for the development of next-generation spintronic devices. Recent studies suggest that the spin Hall effect can switch the topological antiferromagnetic order. However, the switching efficiency remains relatively low. Here, we demonstrate the effective manipulation of antiferromagnetic order in the Weyl semimetal Mn3Sn using orbital torques originating from either metal Mn or oxide CuOx. Although Mn3Sn can convert orbital current to spin current on its own, we find that inserting a heavy metal layer, such as Pt, of appropriate thickness can effectively reduce the critical switching current density by one order of magnitude. In addition, we show that the memristor-like switching behaviour of Mn3Sn can mimic the potentiation and depression processes of a synapse with high linearity—which may be beneficial for constructing accurate artificial neural networks. Our work paves a way for manipulating the topological antiferromagnetic order and may inspire more high-performance antiferromagnetic functional devices.


Supplementary Note 3. Distinguishing orbital torque from spin torque in Cu/CuOx
In main text, we demonstrated deterministic switching in Mn3Sn/Cu/CuOx heterostructure.
Here, to better distinguish the switching driving force from Cu/CuOx, we implemented second harmonics measurement in two samples Ni(5 nm)/Cu/CuOx and Co(5 nm)/Cu/CuOx.The following is our idea for clarifying the source of the torque: If the switching driving force is indeed the orbital torques, then these two samples should show distinct SOT effective fields, because Ni has stronger SOC than Co.However, if the driving force is the spin torques from interfacial Rashba SOC at the CuOx/Cu interface or the bulk spin Hall effect of CuOx as suggested by the reviewer, then these two samples should show similar SOT effective fields.Supplementary Figure 3a shows the measurement setup where we rotate the magnetic field in the xy plane with an angle φ to the current direction.Supplementary Figure 3b and 3c show the first harmonic signal ( ) and second harmonic signal ( ) in Ni/Cu/CuOx sample under H = 0.5 T, respectively.The second harmonics signal can be fitted using the following equation suggested in [Nat. Commun., 12, 7111, 2021].
where Beff, , , and are the effective field, y-polarized damping-like effective field, y-polarized field-like effective field, z-polarized damping-like effective field and z-polarized field-like effective field, respectively.Here, we mainly focus on .
Supplementary Figure 3d  To better quantify the amount of spin torques provided by Pt/Mn bilayer, we implemented spintorque ferromagnetic resonance (ST-FMR) measurement in Co/Pt/Mn trilayer where the Co thickness is fixed to 5 nm.The measurement setup is shown in Supplementary Figure 7a.All the Co/Pt/Mn films were fabricated into 5-μm-wide stripes and connect to two metallic pads.
The applied frequency varies from 3 GHz to 9 GHz.For instance, Supplementary Figure 7b plots the obtained ST-FMR signals Vmix as a function of applied in-plane magnetic field Hex in the sample Co/Pt (2 nm)/Mn (10 nm).The spectrum can be well fitted to the sum of symmetric and anti-symmetric Lorentzian functions, which is written as 2 where Δ, H0, S and A are the linewidth (full width at half maximum), the resonant magnetic field, the symmetric Lorentzian coefficient and the anti-symmetric Lorentzian coefficient, respectively.Then, the effective SHA in the system can be calculated by 2

Supplementary Note 10. Harmonic measurement in Mn3Sn devices 1) First harmonics signal analysis
Regarding the kagome plane in xz plane consists of three Mn atoms with the unit magnetic moment ( , , ) (see illustration in Supplementary Figure 10) and if we consider the spin interaction microscopically, the total energy per unit volume can be expressed by 4 ). ( where and are the exchange and DMI energy coefficients, respectively, is the magnetization of a single sublattice, and 1 is the anisotropy energy coefficient.( , , ) are the single-ion uniaxial anisotropy axes for three sublattices, which correspond to the nearest neighbor Mn-Sn bond direction, and therefore can be written as In general, ≫ ≫ 1, which makes ( , , ) forms 120° angle with respect to each other.The chirality is determined by the DMI.By defining ( , , ) as the angle of ( , , ) in the kagome plane, we have the relation ( , , ) = ( , , ) + ( , , ) .
We then define the cluster magnetic octupole moment by = + − + , where is the rotation matrix, ± rotate ] with a factor Δ to modify the change of .Then, we consider a straininduced uniaxial magneto-crystalline anisotropy K2 on all sublattices, with the corresponding easy axes are all along the direction ( ).Then, the new microscopic magnetic energy can be written as 4 : ). (5) Here we assume that the external field is applied within the kagome plane, then one can rewrite the magnetic energy as 4 : The equilibrium condition is now = = = 0. Note that an analytical solution for the equilibrium angle is too complicated to derive, one can determine ( , , ) numerically, which further leads to under any applied field condition.Using literature reported values of 0 0 = 0.65 T, 1 = 6.3 × 10 5 J/m 3 , = 1.0 × 10 8 J/m 3 and substituting into the anomalous Hall resistance Rω expression = , we find that the model fits the ( ) data very well (see Fig. 3e), where the only two fitting parameters are and Δ .

2) Second harmonics signal analysis
We assume that the SOT applies onto each sublattice separately and the applied field and ( , , ) remain in the kagome plane under zero applied current.Then, the magnetic dynamics can be described with the coupled Landau-Lifshitz-Gilbert equations: Previously we defined = + − + by rotation and mirror reflection transformations on ( , , ) .We would like to write out the equation that governs the dynamics of : where the first and second terms are the torque from effective magnetic field and dampinglike (DL) spin torque .Here we dropped the Gilbert damping torque for simplicity.In the dynamical switching process where ≠ , spin torque can lead to a small out-of-kagomeplane tilt of ( , , ) and induces fast precession of ( , , ) and .Since we use a lowfrequency ac current, we here adopt the quasi-static limit of ( , , ) ≈ and ≈ .
According to the handedness anomaly in Ref.
In static case, the net torque on the octupole moment should be zero + = 0, which leads to Here we adopt the macroscopic anisotropy energy coefficient , in stead of the microscopic individual spin anisotropy parameter Δ and .Consequently, the second harmonic resistance can be obtained as = ∆ ( ) where is the peak ac current.
With , obtained in the first harmonic measurement, we also well fit our curve in Fig. 3f, with the only fitting parameter being HDL.The effective spin Hall angle can then be calculated by = ( ) ℏ , where , t, ℏ and JSOT are the magnetization of a sublattice moment, the Mn3Sn thickness, the reduced Planck constant and the average current density in the source layer, respectively.
Ms, t, d and 4 are the saturated net magnetization of Co, the thickness of Co, the thickness of Pt/Mn bilayer and the demagnetization field of Co, respectively.Supplementary Figure 7. a, ST-FMR measurement setup.b, ST-FMR signals versus Hex under different frequency.
that our grown Mn3Sn is polycrystalline, we should consider extra magnetic field anisotropy terms brought by lattice constant changes, etc.In most case, the lattice constant changes in the in-plane direction, thus, the exchange constant is going to be different from and .Therefore, we change the exchange energy term to [(1